Questions tagged [jordan-normal-form]

The Jordan normal form of a matrix is a similar block matrix having diagonal blocks when the matrix is diagonalizable and diagonal + nilpotent blocks more generally.

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Show The Jordan Normal Form Of $\varphi$.

Fix a nonnegative integer $n$, and consider the linear space $$\mathbb{R}_n\left [x,y \right ] := \left\{ \sum_{\substack{ i_1,i_2;\\ i_1+i_2\leq n}}a_{i_1i_2}x^{i_1}y^{i_2}\quad\Big|{}_{\quad}a_{...
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Find a flag to transform a matrix to an upper triangular one

Consider $F: \mathbb{R^3} \to \mathbb{R^3}$ represented by: $ A= \begin{bmatrix} 1 & 1 & 2 \\ -2 & 5 & 6 \\ 1 & -2 & -2 \\ \end{bmatrix} $ , eigenvalues: $...
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874 views

An inverse of Jordan matrix - basis

Let $A\in M_{n\times n}$ be and invertible matrix over complex field and we assume it's already at Jordan form where $B=\{v_1,…,v_n \}$ is Jordan basis for A. Find Jordan form and Jordan basis for $...
6
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1answer
242 views

An explanation for a Jordan normal form proof from the Kaye and Wilson book

In this proof of Jordan normal form in the Kaye and Wilson book, then for a transformation $T$ with minimal polynomial $m(x) = (x-e)^k$, they take a basis of $\texttt{ker}\;T$, extend it to a basis of ...
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2answers
49 views

Have I found the Jordan form correctly?

I am given that the minimal polynomial and characteristic polynomial of a matrix are both $(x-1)^2(x+1)^2$. I have found the Jordan form to be $$\begin{bmatrix}1&1&0&0\\0&1&0&0\...
5
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1answer
71 views

Can basis of kernel be extended to a Jordan basis?

Let $A\in\mathbb C^{n\times n}$ be nilpotent. A Jordan basis of $A$ is a basis of $\mathbb C^n$ with respect to which $A$ has Jordan normal form. Assume that we do not know the Jordan structure of $A$....
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192 views

Lost on rational and Jordan forms

I'm having a lot of trouble trying to understand rational canonical form, primary rational canonical form, and Jordan form. I've looked at the posts about this, but I haven't been able to understand ...
5
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2answers
697 views

Generalization of the Jordan form for infinite matrices

Under what conditions is it the case that for a matrix $M$ whose rows and columns are indexed by a countably infinite set $S$ one has a Hamel basis consisting of generalized eigenvectors (i.e. $v \in \...
5
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2answers
1k views

Jordan form exercise

What am I doing wrong? I've been learning how to put matrices into Jordan canonical form and it was going fine until I encountered this $4 \times 4$ matrix: $A=\begin{bmatrix} 2 & 2 & 0 &...
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155 views

Jordan Block of Kronecker Product

Let $A$ be a $(p\times p$)-Jordan block of generalized eigenvalue $\lambda$. Let $B$ be a $(q\times q$)-Jordan block of generalized eigenvalue $\mu$. Then what is the Jordan canonical form for $A\...
4
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1answer
178 views

Why block-diagonal form for nilpotent matrices?

I am currently reading Jim Hefferon's Linear Algebra. In chapter 5, nilpotence, strings, he goes through the process of finding a string basis of a map, and proves that there exists a string basis ...
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1k views

Prove the direct sum of generalized eigenspaces is the whole vector space

Given a $n\times n$ matrix $A$ over an algebraically closed field, let $\lambda_1,...,\lambda_k$ be its eigenvalues, and let $V_{\lambda_i}$ be the generalized eigenspace of $\lambda_i$. The question ...
4
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1answer
344 views

relation between minimal polynomial and jordan normal form

I just solved some exercises on minimal polynomials and i remember that there is a relation between the minimal polynomial and the jordan normal form. But my question is the following : knowing the ...
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44 views

minimal polynomial of a matrix B given minimal polynomial of $B^2$

If we are given a minimal polynomial for a matrix $B^2$ can we deduce the minimal polynomial for $B$ $?$ Example: if the minimal polynomial for $B^2$ is $m(\lambda) = \lambda^4$ then can we deduce ...
3
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1answer
130 views

Using Jordan Normal Form to determine when characteristic and minimal polynomials are identical

Say I want to immediately write down a matrix with an identical minimal and characteristic polynomial. Say, $$ (t-1)^{3}(t-2). $$ My first instinct is to write down Jordan Blocks in a block ...
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59 views

Find the Jordan Canonical Form of the given transformation

The transformation here is $T(f(x)) = f(x + 1) + f(x − 1)$ which is a linear endomorphism on V, where $V={f(x) ∈ R[x] : deg f(x) ≤ 2017}$ So I have to find the jcf J of T. Along with a basis of B. $...
3
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1answer
268 views

Minimal polynomial and possible Jordan forms

Let $A$ be an $8\times 8$ complex matrix with characteristic polynomial $$p_A(x)=(x-1)^4(x+2)^2(x^2+1)$$ and minimal polynomial $$m_A(x)=(x-1)^2(x+2)^2(x^2+1).$$ Determine all possible Jordan ...
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153 views

Classification of bilinear forms: operator $A^{-1} A^T$ for bilinear form $A$

I would like to understand a classification of non-degenerate (not necessary symmetric or skew-symmetric) bilinear forms over an algebraically closed field via an operator $\kappa=A^{-1} A^T$ for a ...
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Differential Equations: Jordan Form of a Matrix

I am using Lawrence Perko's book Differential Equations and Dynamical Systems, for my Differential Equations course. At the moment we are going over Jordan Forms of a linear system $x^{'}(t) = Ax$, ...
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341 views

Jordan normal form theorem proof question

Theorem: Assume that the characteristic polynomial $x_f$ splits into linear factors. Then there exists a Jordan normal form for f. The Jordan normal form is unique up to the order of the Jordan blocks....
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39 views

Jordan basis of $\mathcal{M}_{\mathcal{T}}(A)$

Let $A\in M_{n\times n}(\mathbb{R})$ be a matrix. Let $\mathcal{B}$ be a basis of $\mathbb{R}^n$ and $X:=\mathcal{M}_{\mathcal{B}}(A)$. If $\mathcal{S}$ is the basis for which $\mathcal{M}_{\mathcal{S}...
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1answer
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Basics of Jordan matrix, please clarify the following

Let $$J=\oplus_{i=1}^{k} J_{n_i}(\lambda)$$ where $J_{n_i}(\lambda)$ is a jordan block of size $n_i$ with $\lambda $ on its diagonal, and $\sum_{i=1}^{k}n_i = n $, so $J$ is $n\times n$ matrix, ...
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How many similarity classes are zeroes of a given polynomial.

I am looking for an easy way of calculating the number of similarity classes of complex matrices that satisfy some polynomial $p(t)$. As an example consider $p(t)=(t-1)^3(t+1)^4$ and $5\times 5$ ...
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40 views

Analogy of Jordon Normal Form for Antilinear Maps

Given complex vector spaces $V$, and antilinear $T:V \rightarrow V$, then if we fix a basis of $V$, we can represent $T$ by the matrix of the linear $T \circ J$, where $J$ is complex conjugation. I ...
2
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1answer
55 views

How to find a Jordan basis and a Jordan matrix for a nilpotent matrix?

I am trying to find a general step-by-step "easy" / "intuitive" solution to finding Jordan basis and Jordan matrix (based on the basis) for a nilpotent matrix. If you can add an intuition for the ...
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45 views

How can I find the Jordan form of this upper triangular Toeplitz matrix?

Given an $n \times n$ matrix $A$ whose $(i,j)$ entry is $$a_{ij} = \begin{cases} n-j+i & \text{if } j \geq i\\ 0 & \text{otherwise}\end{cases}$$ find its Jordan form. I know that ...
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For which values ​the matrix is ​diagonalizable

For which values ​​of $a$ matrix $A$ is ​​diagonalizable? $$A = \pmatrix{0&i\\i&a}$$ in the case that it is not diagonalizable determine a base of Jordan Attempt: The minimal polynomial ...
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1answer
127 views

Linear Algebra : Jordan Canonical form (Jordan blocks and the Super-Diagonal)

In terms of Jordan Canonical Form, and more specifically about Jordan Blocks. When there is a definition about Jordan Blocks they say the eigenvalues go on the principle diagonal and the diagonal ...
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When is possible to use an orthogonal matrix to put in Jordan form a matrix?

I know that if I have a symmetrical matrix defined on $R$, it is always diagonalisable and I can always find beetwen the matrix of its eigenspaces an orthogonal matrix. While if I have a non ...
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67 views

Trouble in proof of Jordan Canonical Theorem

Jordan Canonical Theorem stated that: Let $K$ be an algebra closed field. Let $V$ be a nonzero, finite dimensional vector space over $K$, and let $\psi \in \operatorname{End}_K(V)$. Then there ...
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53 views

Classification of Matrices and normal forms

As is shown in couses of Linear Algebra, for every square matrix $A$ one can choose $S,T,P\in GL(n,K)$ so that $SAT^{-1}=\operatorname{diag}(1,...1,0,...,0)$ and $PAP^{-1}$ is in Jordan normal form. ...
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36 views

Polar Decomposition in Real Algebraic Groups

Every element $g \in GL(n,\mathbb{C})$ has a unique Jordan decomposition $$ g = g_u g_s $$ where $g_u$ is unipotent, $g_s$ is semisimple (i.e. diagonalizable over $\mathbb{C})$ and $g_ug_s=g_sg_u$. It ...
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Calculate $\det(p(A))$

Let $n \in \mathbb N$, $A \in \mathbb C^{n \times n}$ be nilpotent and $l\in \mathbb N$. Further, let $$p = \sum_{i=0}^n \alpha_i A^i \in \mathbb C[t]$$ be a polynomial. Show that zero is the only ...
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143 views

Problem on characteristic polynomial and minimal polynomial

I am given two matrices a and b such that characteristic polynomial and minimal polynomial of a and b are equal I have to check If they are similar JC form of a and b are same Rank(a) = rank(b) If ...
2
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1answer
336 views

Jordan form, number of blocks.

Suppose I have an eigenvalue $\lambda$, now I want to determine the number of Jordan blocks corresponding to that eigenvalue, as well as size of each block. I know that: 1) number of blocks is equal ...
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80 views

Linear transformation with diagonalizable power

Let $f: \mathbb{C}^5 \to \mathbb{C}^5 $ a linear transformation such that $f^3$ is diagonalizable, but $f^2$ is it not. Is it true that $f$ necessarily has a $3 \times 3$ jordan block with a null ...
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Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley decomposition of A

Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley. I got a hint which says, that I should write $A^p-I$ as a power of some other matrix....
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109 views

If all eigenvalues are < 1, fixed point iterations converges to the only solution

Theorem states that for every initial value fixed point iteration x = Bx+b converges to the only solution of the system if all $|\lambda| $ < 1. Prove it using Jordans normal form. Initial form is ...
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1answer
549 views

Nilpotent Matrix is Similar to a block diagonal matrix

Prove that any nilpotent matrix is similar to a block diagonal matrix whose blocks are matrices with 1's along the first super diagonal and 0's elsewhere. I'm not sure where to start exactly. Any ...
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355 views

Show the form of $J$ and $P$ for Leslie matrix $A$ when $A = PJP^{-1}$

I'm trying to solve this for a homework assignment. The Jordan Normal form theorem states that every complex $n \times n$ matrix $A$ van be written as $A=PJP^{-1}$, where $J$ is the diagonal matrix ...
2
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1answer
73 views

Describing the space of matrices which “jordanize” a given matrix

This is a naive linear algebra question. I apologize for the level but I could not find an answer in the literature. Let $A$ be a $n$ by $n$ matrix (say over $\mathbb C$). Suppose the Jordan form of $...
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0answers
41 views

Show that jordan matrix with k blocks in diagonal has exactly k independent eigenvectors?

How can I prove that a Jordan matrix with k blocks in the diagonal has exactly k independent eigenvectors. Can you help me to find a formal proof of this statement?
2
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1answer
149 views

Examples of decomposable nilpotent endomorphisms

Every body knows that if $V$ is the real vector space $\mathbb R[X]_{\leq n}$ of polynomials of degree at most $n$, then the linear map $$D:p\in V\mapsto p'\in V$$ given by derivation is nilpotent. It ...
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1answer
39 views

Direct sums of invariant subspaces

Let $A$ be a complex $n\times n$ matrix, with its Jordan carnonical form as $J=diag(J_1,\cdots,J_s)$. Then there exists an invertible matrix $P$ such that $P^{-1}AP=J$. It is easy to verify that $\Bbb ...
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350 views

Jordan canonical form when field is not algebraically closed

Suppose we have a linear operator $T : V \to V$, where $V$ is a vector space $V$ over a field $F$. Now if $F$ is not algebraically closed, we don't necessarily know that $T$ has a Jordan canonical ...
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172 views

Jordan Normal Form of a self-adjoint Linear Transformation

Let $V$ a finite inner product space, $dim V = n \geq 3$. Let $w_1,w_2 \in V$ such that: $<w_1,w_2>=0$, $||w_1||=||w_2||=1$ where $||w||$ is the norm of a vector $w$. The inner product ...
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1answer
76 views

If $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, find $t$ operators such that $T=T_1\oplus\dots\oplus T_t$

Be $T\in \mathscr{L}(V)$ a linear operator with characteristic polynomial $p_T(x)=(x-\lambda_1)^{n_1}\dots(x-\lambda_t)^{n_t}$, $n_i\geq 1$ and $\lambda_i\neq\lambda_j$ if $i\neq j$. Show that $T$ can ...
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550 views

Complex eigenvalues Jordan real matrix

As I posted here and here I'm studying Jordan forms and similar concepts. I've got a problem with complex eigenvalues in jordan real matrices. I know (at least I think so) how to compute the Jordan ...
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82 views

What does power of a factor in a minimal polynomial mean in rational form?

Let $V$ be a finite dimensional vector space and $T$ be some linear operator. Suppose the minimal polynomial has a factor $(x-c)^2$. If $T$ has Jordan form, then we can assert that the biggest jordan ...
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52 views

Lipschitz continuity of invariant subspaces for parametrized matrices

Let $A(t)$ be a one-dimensional parametrized family of linear operators on $\mathbb{R}^m$ that has smooth dependence on $t$. Let $V_0\subset \mathbb{R}^n$ be an $n$-dimensional invariant subspace for ...